g ρ dx dy dz,
ρ being the density of the mass, and dx, dy, dz the three dimensions of an infinitesimal portion of it.
Now, since we know that gravity in the unit of time imparts a finite velocity to every point of matter in the atom, we must admit that the action exerted on the infinitesimal mass ρ dx dy dz has a finite value; and therefore, since the volume dx dy dz is an infinitesimal of the third degree, the density ρ must be an infinite of the third order. But a continuous mass whose elements have an infinite density has itself an infinite density; hence, if its volume has finite dimensions, the mass itself (which is the product of the volume into the density) is necessarily infinite, and will have an infinite weight. Hence the assumption of continuous matter leads to an absurdity. The assumption is therefore to be rejected as evidently false.
We will put an end to the series of our proofs by pointing out the intrinsic and radical reason why matter cannot be continuous. The matter which is under the form is a potency in the same order of reality in which its form is an act. Now, the only property of a potency is to be liable to receive some determinations of a certain kind; and the property of a potency whose form is an active principle of local motion must consist in its being liable to receive a determination to local movement. Hence, as the matter receives its first being by a form of a spherical character, and becomes the real central point from which the actions of the substance proceed, so also the same matter, when already actuated by its essential form, receives any accidental determination to local movement; and, inasmuch as it is liable to local movement, it is in potency to extend through space—that is, to describe in space a continuous line; and when it actually moves, it actually traces a continuous line—that is, it extends from place to place, continuously indeed, but successively; whence it is manifest [pg 494] that its extension is nothing but Actus existentis in potentia ut in potentia Aristotle would say, viz., an actual passage from one potential state to another. Such is the only extension of which matter is capable. Such an extension is always in fieri, never in facto esse; always dynamical, never statical; always potential and successive, never formal or simultaneous. We can, therefore, ascribe to matter potential continuity, just as we ascribe to its active principle a virtual continuity; for the passivity of the matter and the activity of the form correspond to one another as properties of one and the same essence; and whatever can be predicated actively or virtually of a substance on account of its form can be predicated passively or potentially of the same substance on account of its matter.
These remarks form a complement to our fifth argument, where we proved that no substantial and no accidental act could make matter actually continuous. For, since matter cannot receive any accidental act, except the determination to local movement, and since this movement, although continuous, is essentially successive, it follows that by such a determination no actual and permanent continuity can arise, but a mere continuation of local changes. Thus matter, according to its potential nature, has only a potential extension; or, in other terms, it is not in itself actually continuous, but is simply ready to extend through space by continuous movement.
The preceding proofs seem quite sufficient, and more than sufficient, to uproot the prejudice in favor of material continuity; we must, however, defend them from the attacks of our opponents, that no reasonable doubt may remain as to the cogency of our demonstration.
First objection.—The globe and the plane, of which we have spoken in our third argument, though destitute of proportional parts suitable for a statical contact, become proportionate to one another, says Goudin, by the very movement of the one upon the other; and thus our third argument would fall to the ground. For a successive contact partakes of the nature of successive beings. Hence, as time, although having no present, except an indivisible instant, becomes, through its flowing, extended into continuous parts, so also the contact of the globe with the plane, although limited to an indivisible point, can nevertheless, by its flowing, become extended so as to correspond to the extended parts of the plane. For, according to mathematicians, a point, though indivisible when at rest, can by moving describe a divisible line.
To this we answer that a globe and a plane cannot by the movement of the one on the other acquire proportionate parts. For, although it is true that a successive contact partakes of the nature of the successive being which we call movement, it is plain that it does not partake of the nature of matter. In fact, the material plane is not supposed to become continuous through the movement of the globe, but is hypothetically assumed to be continuous before the movement, and even before the existence, of the said globe. The continuous movement is, of course, proportionate to a continuous plane; but it is evident that it cannot originate any proportion between the plane and the globe; because this would be against the essence of both. No part of the plane can be spherical, [pg 495] and no part of the globe can be plane; hence, whatever may be the movement of the one upon the other, they will never touch one another, except in a single point.
That time, although having no present, except an indivisible point, becomes extended by flowing on, is perfectly true; but this proves nothing. For, in the same manner as the act of flowing, by which time flows, has nothing actual but a single indivisible instant, so also the act of flowing, by which the contact of the globe with the plane flows, has no actuality but in an indivisible point of space; and as an indivisible instant by its flowing draws a line of time without ever becoming extended in itself, so also an indivisible point by its flowing draws a line in space without ever becoming extended in itself; and as the instant of time never becomes proportionate to any finite length of time, so also the point of contact never becomes proportionate to any finite line in space.
That a line, therefore, arises from the flowing of a point in the same manner as time from the flowing of an instant, is a plain truth, and there was no need of Goudin's argumentation to make it acceptable. To defeat our argument, he should have proved that the actual flowing of an instant takes up a length of time. If this could have been proved, it would have been easy to conclude that the flowing contact also extends through a length of space. But the author did not attempt to show that an instant of time flows through finite lengths of time. It is evident, on the contrary, that an instant flows through mere instants immediately following one another. And thus the objection has no weight.